Characterizations of complex symmetric Toeplitz operators

Abstract

We characterize Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is S-Toeplitz for some unilateral shift S and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift S. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results are related to a question raised by K. Guo and S. Zhu.